Around 25 posts could go at Royal Avenue if the store closes. It is the clothing chain's only outlet in Northern Ireland.
Across the UK, 2,500 jobs are at risk across over 121 stores.
Administrator Ernst & Young said the shops would continue to trade while it looks for a buyer for the business.
Owned by private equity firm TPG, the clothing chain started in Leeds as a men's denim retailer in 1986 and now sells mens and womenswear brands including G-Star, Diesel and Firetrap.
TPG bought Republic in June 2010 in a deal worth around £300 million, with the founders understood to have retained a significant stake.
Ernst & Young said it had made 150 staff redundant at Republic's headquarters in Leeds.
Administrator Hunter Kelly said: "Republic suffered poor trading results in the autumn, and whilst sales picked up in December there has been a sudden and rapid decline in sales in late January.
He added: "We will continue to trade Republic, with a view to selling the business as a going concern. The brand Republic is well recognised, particularly in the North. It has a powerful website offering, owns well-known brand names, and has some very attractive and profitable stores."
Chairman Andy Bond, a former Asda boss, is said to have stepped down last week after less than two years.
The chain recently hired ex-TK Maxx boss Paul Sweetenham as chief executive.